Syllabus  -^e    in 

)Tnetrty   of 
ree    DJ  Ions 


bv 


Dr. 

Gift   of 
and  Mrs.   A.B. 

Pier 

SYLLABUS 


RSE  IN  ANALYTICAL  GEOMETRY 


BOSTON: 

PUBLISHED  BY   GINN,    HEATH 


)PTBIGHTED    BV    GlKST,    HEATH 


MATHEMATICS. 


THE  following  volumes  in  Wentvvorth's  Mathematical 
Series  arc  now  ready  for  delivery :  — 

Elements  of  Algebra  .     9|  ^B  ^1 |H  $1.12 

Complete  Algebra  ...  1.40 

Plane  Geometry     ^Hj  ^B  ^H II  ^1  '^ 

Plane  and  Solid  Geometry  .     ^H  1-25 

Plane  and  Solid  Geometry,  and  Plane  Trigonometry.         .         1.40 

Plane  Trigonometry.     Paper 

Plane  Trigonometry  and  Tables.     Paper        .     ^^1     •  *^ 

Plane  and  Spherical  Trigonometry     JJHJ  ^| 

Plane  and  Spherical  Trigonometry,  Surveying,  and  Navk 

Plane  and  Spherical  Trigonometry,  and  Surveying. 

Tables  ^B  HBH  ^B  ••  •B  ^H  1  •  25 

Surveying.  .  P:j_pr- 

Trigonometric  Formulas  (Two  Charts,  each  30  X  40  i:;  1.00 

Weutworth  &  Hill's  Five-Place  Logarithmic  and  Trl 

nometric  Tables.     (Seven  Tai.i.  H^ffilM[ 

Wentworth  &  Hill's  Five-Place  Logarithmic  and  Tri: 

nometric  Tables.    Complete  Edition        .  1.00 

Wentworth  &  Hill's  Practical  Arithmetic        .     BK|  1-00 

Wentworth  &  Hill's  Examination  Manual.    I.    Arithmetic 
Wentworth  &  Hill's  Examination  Manual.    II.   Algebra  . 
Wentworth  &  Hill's  Exercise  Manual.    II.    Algebra 
(The  last  two  may 'be  had  in  one  vein 


ANNOUNCEMENTS. 

EXEBCISE  MANUAL  OF  AEITHMETI 
iCISE  MANUAL  OF  GEOMETRY. 

McLEtLAN's  UXIVEIiSF!  •  BRA. 

WENTWORTII'S  GRAMMAR  SCHOOL  ARJTJIMET. 
YOUTH'S  PRIMARY  SCHOOL  ARTTTLM1 

GINN,  HEATH,  &  CO.,  Publishers. 

BOSTON,   N 


SYLLABUS 

OF   A 

COURSE  IN  ANALYTIC  GEOMETRY  OF  THREE 
DIMENSIONS. 


1.    Explain  the  method  of  denoting  the  position  of  a  point  in 
space  by  Cartesian  Coordinates. 

State  the  convention  concerning  the  signs  of  coordinates. 

•v 

>\          2.    State  the  convention  concerning  positive  rotation  about 
r^       any  axis. 

3.    Explain  Polar  Coordinates  in  space. 
Obtain  formulas  for  transforming  from  rectangular  to  polar 
K*     coordinates. 


[1]  2/  =  rsin<£  cos0, 

z  =  r  sin  <f>  sin  0. 

4.    PROBLEM.      To   find   the   distance   between    two    points 
whose  coordinates  are  given. 


^          [2]  D  = 


\i          5.    PROBLEM.     To  divide  a  line  in  any  given  ratio,  m-i :  m2. 

f31     x  = ,   v  = 

nia  +  nil 

If  the  line  is  bisected, 


4G2G19 


6.  Define  the  projection  of  a  point  on  a  plane  ;  of  a  line  on 
a  plane ;  of  a  point  on  a  line ;  of  a  line  on  a  line. 

Prove  that  the  projection  of  a  line  I  on  a  line  p  is 

[5]  lp  =  I  cos  a, 

where  a  is  the  angle  between  the  two  lines. 

7.  Show  that  the  rectangular  coordinates  of  any  point  are 
the  projections  of  the  radius  vector  of  the  point  on  the  three 
axes ;  so  that  if  a,  (3,  and  y  are  the  angles  made  by  r  with  the 
axes  of  X,  T",  and  Z  respectively,  we  have 

[6]  #  =  rcosa,   y  =  rcos{3,   z  = 

and 

[7]  r2  = 

a,  /?,  and  7  are  called  the  Direction  Angles  of  r. 

8.  Define  the  Direction  Cosines  of  a  line,  and  prove  that  the 
sum  of  their  squares  is  unity. 

[8]  COS2a  +  COS2  ft  +  COS2y  =  1 . 

The  radius  vector  of  a  point  and  its  direction  angles  may  be 
used  as  a  set  of  polar  coordinates. 

9.  PROBLEM.     To  find  the  angle   between  two  lines   when 
their  direction  cosines  are  given. 

[9]      COS  6  =  COS  aa  COS  a2  -}-  COS  /?!  COS  @2  +  COS  yj  COS  y2. 

The  lines  are  perpendicular  if 

[10]      0  =  COStt!  COSa2  +  COS/?!  COS/?2  +  COSyi  COSy2. 

They  are  parallel  if 
[11]  a1  =  a2,     /81  =  /?2,     71  =  72- 


TRANSFORMATION  OF  COORDINATES. 

10.   PROBLEM.     To  transform  to  a  new  set  of  axes  parallel  to 
the  old. 

[12]       x  = 


11.   To  transform  from  one  set  of  axes  to  a  second  set  hav- 
ing the  same  origin. 

x  =  x'  cos  aj  +  y'  cos  a2  +  z'  cos  03, 
[13]  y  = 

Z  = 


Show  that  the  degree  of  an  equation  cannot  be  altered  by 
either  of  these  transformations. 

12.  Explain  what  is  meant  in  space  by  the  locus  of  an  equa- 
tion or  pair  of  equations. 

Show  that  a  single  equation  between  cc,  y,  and  z  represents  a 
surface.  That  a  pair  of  such  equations  represent  a  line. 

Show  how  the  form  of  a  surface  whose  equation  is  given  may 
be  investigated  by  means  of  its  plane  sections. 

An  equation  containing  only  two  variables  represents  a  cylin- 
drical surface. 

Show  how  to  obtain  the  equation  of  a  surface  formed  by  re- 
volving a  plane  curve  about  one  of  the  axes. 

THE  PLANE. 

13.  PROBLEM.     To  find  the  equation  of  a  plane  in  terms  of 
the  perpendicular  from  the  origin  and  its  direction  cosines. 

[14]  aJCOSa  +  2/COS/J  +  ZCOSy  =  p. 


4 
14.   Prove  that  every  equation  of  the  first  degree, 


represents  a  plane,  and  show  how  to  reduce  it  to  the  form  [14]. 

15.    Express  the  equation  of  a  plane  in  terms  of  its  intercepts 
on  the  axes. 


16.  PROBLEM.     To  find  the  equation   of   a  plane   through 
three  given  points. 

17.  PROBLEM.     To  find  the  distance  from  a  given  point  to  a 
given  plane. 

[16]  D  =  XiCOSa  +  ylCOS(3  +  Z1COSy  —  p. 


18.  PROBLEM.     To  find  the  angle  between  two  planes. 
[17]     cos  6  =  A1A2-{-B1B2  +  O1 


They  are  perpendicular  if 

[18]  A,A2  +  B,B2  +  0,  C2  =  0. 

They  are  parallel  if 

[19"|  S  —  •  fS  —s  Jr3. 

U3.2         X>2         ^2 

19.  PROBLEM.  To  find  the  equation  of  a  plane  passing 
through  a  given  point  and  parallel  to  a  given  plane  ;  passing 
through  two  given  points  and  perpendicular  to  a  given  plane. 


THE  STRAIGHT  LINE. 

20.  Show  that  the  equations  of  a  line  may  always  be  thrown 
into  the  form 

„_„,+«, 

y  =  nz  +  b. 

21.  Find  the  equations  of  a  line  in  terms  of  its  direction 
cosines,  and  the  coordinates  of  a  point  through  which  it  passes. 

[21] 


COS  a          COS/3         COSy 

22.  PROBLEM.     To  throw  the  equations  of  any  line  into  the 
form  [21] 

x—  a  y  —  b  z—c 

[22]  m  n  I 

Vl  +  m2  +  n2      Vl  +  m2  +  n2      Vl  +  m2  -f-  w2 

23.  PROBLEM.      To   find   the    equations  of  a   line  passing 
through  two  given  points. 

as  —  Xi         —          z  —  Z 


[23] 


2/2  - 


24.  Problems  concerning  the  angles  between  lines,  or  between 
lines  and  planes,  can  be  readily  solved  by  the  use  of  the  direc- 
tion cosines  of  the  lines  and  those  of  the  normals  to  the  planes. 

THE  SPHERE. 

25.  PROBLEM.     To  find  the  equation  of  a  sphere  in  terms  of 
the  coordinates  of  its  centre  and  the  length  of  its  radius. 

[24]  (x-o)t  +  (y-b)a  +  (z-c)a=ia. 

When  the  centre  is  at  the  origin,  this  becomes 
[25] 


26.  Show  that  the  most  general  form  of  the  equation  of  a 
sphere  is 

[26]       tf  +  yt  +  zt  +  Gx  +  Hy+lz  +  K^Q, 

and  show  how  to  reduce  any  equation  of  this  form  to  the  form 
[24],  and  thus  to  determine  its  centre  and  radius. 

27.  PROBLEM.     To  find  the  equation  of  a   sphere  passing 
through  four  given  points. 

28.  Show  that  any  two  spheres  intersect  in  a  circle. 

29.  Find  the  equation  of  the  tangent  plane  at  a  given  point 
on  the  surface  of  the  sphere,  x2  +  y2  +  «2  =  1s- 

[27] 


30.  PROBLEM.     To  find  the  equations  of  the  normal  at  any 
point  of  the  sphere. 

[28]  *  =  */  =  *. 

x,     ^     Zi 

Prove  that  every  normal  is  a  radius. 

31.  PROBLEM.     To  find  the  locus  of  points  dividing  harmoni- 
cally secants  drawn  from  a  given  point  to  a  sphere. 

[29]  Xt  x  +  yly  +  zl  z  =  ri. 

This  is  called  the  polar  plane  of  the  given  point,  and  passes 
through  the  points  of  contact  of  all  the  tangents  that  can  be 
drawn  from  the  given  point  to  the  sphere. 

32.  Prove  that  if  several  points  lie  in  a  plane,  their  polar 
planes  pass  through  the  pole  of  the  given  plane  ;    and  con- 
versely, that  if  several  planes  pass  through  a  point,  their  poles 
lie  on  the  polar  plane  of  that  point. 


33.  Prove  that  the  polar  plane  of  a  point  is  perpendicular  to 
the  line  joining  the  point  with  the  centre  of  the  sphere,  and  that 
the  product  of  the  distance  of  the  pole  from  the  centre  and  the 
distance  of  the  polar  plane  from  the  centre  is  equal  to  the 
square  of  the  radius. 

34.  PROBLEM.     To  find  the  locus  of  the  middle  points  of  a 
set  of  parallel  chords. 

[30]  X  COSa  +  y  COS/3  +  2!  COSy  =  0. 

Such  a  locus  is  a  diametral  plane. 
Define  diameter;  conjugate  diameters. 


THE  CENTRAL  QUADRICS. 

35.  The  central  quadrics  are  the  ellipsoid^  the  bi-parted  Jiyper- 
boloid,  the  un-parted  hyperboloidj  and  the  cone. 

q&  qj2  g2 

Investigate  their  forms. 

36.  Find  the   equation  of  the   tangent   plane   to   a   central 
quadric. 

T321  ^i*^  .  y\y  .  ^i_2    i 

a?        b2      ~c^ 
Of  the  normal  line. 

[33]  —  (x  —  #,)  =  —  (y  —  V,)  =  —  (z  —  z,}. 

"-          J  /v.      >•  *•'  ni     ^"  •**•/  ~      \  ~1/' 


4G2619 


8 

37.   Find  the  equation  of  the  polar  plane  of  a  point  with 
respect  to  a  central  quadric,. 


and  prove  that  sections  31  and  32  apply  to  any  central  quadric 
as  well  as  to  the  sphere. 

38.  Find  the  equation  of  the  diametral  plane  conjugate  to  a 
given  chord  of  a  central  quadric. 

,-Q^-,  iccosa  ,  ycosB  ,  zcosy     n 

L"°J  5 — I Ts!     "I "2     —  u* 

a2  W  c2 

39.  The  diametral  plane  conjugate  to  the  diameter  through 

r  o  /?  T  \      v  i  y      i      /\ 

L  J  ~w+'~w+~^;=^' 

woo 

40.  Show  that  when  two  diameters  are  conjugate,  their  direc- 
tion cosines  are  connected  by  the  relation 

pqr--i    COSctj  COSa2   ,    COS/?!  COS  (32   .   COSyi  COSy2 A 

L"     J      "  o  To  "1  o 


41.    Show  that  the  coordinates  of  any  point  of  a  central 
quadric  can  be  expressed  in  the  form 


[38]         # 
where  X,  /*.,  and  v  are  the  direction  angles  of  an  auxiliary  line. 

42.  Show  that  if  two  diameters  are  conjugate,  the  auxiliary 
lines  corresponding  to  their  extremities  are  mutually  perpendic- 
ular. 


43.  Show  that  the  sum  of  the  squares  of  three  conjugate 
diameters  is  constant. 

44.  Prove  that  the  paYallelopiped  whose  edges  are  three  con- 
jugate diameters  has  a  constant  volume. 

CIRCULAR  SECTIONS. 

45.  Prove  that  through  the  centre  of  every  central  quadric 
two  planes  can  be  drawn,  each  of  which  will  cut  the  quadric  in 
a  circle. 

46.  Show  that  every  plane  section  of  a  quadric  parallel  to  a 
circular  section  is  a  circle. 

47.  Define  the  umbilics  of  an  ellipsoid,  and  show  how  to  find 
them. 

48.  Show  that  the  circular  sections  of  an  hyperboloid  and  of 
its  asymptotic  cone  are  the  same. 

RULED  SURFACES. 

49.  Show  that  on  the  un-parted  hyperboloid  two  sets  of  right 
lines  can  be  drawn,  lying  wholly  in  the  surface  of  the  hyperbo- 
loid ;  and  that  through  every  point  of  the  hyperboloid  one  line 
of  each  set  will  pass. 

50.  Prove  that  the  two  lines  passing  through  a  given  point 
of  the  hyperboloid  lie  in  the  tangent  plane  drawn  at  the  point 
in  question. 

51.  Show  that  each  line  of  one  system  meets  all  the  lines  of 
the  other  system,  and  none  of  the  lines  of  its  own  system. 


10 

52.  Prove  that  an  un-parted  hyperboloid  may  be  generated  by 
the  motion  of  a  line  which  always  touches  three  given  lines,  no 
two  of  which  are  in  the  same  plane. 

53.  Show  that  if  a  line  revolve  about  another  line  not  in  the 
same  plane,  it  will  generate  a  ruled  hyperboloid. 

54.  Investigate  the  properties  of  the  ruled  paraboloid. 

W.  E.   BYERLY, 

Professor  of  Mathematics  in  Harvard  University. 


J.  S.  GUSHING  &  Co.,  PRINTERS,  BOSTON. 


Books  on  English  Literature. 


Aruo: 


Church 


Hurrisou  & 


Hudson  &  Lamb 

Hunt   ..... 
Lambert    ... 


Lounsbury 
Mintc 

Sprugue 


Thorn 

Yonge 


History  Topics    ........... 

English  Literature  ...... 

LnglQ-Saxou  Grammar.    -  ..... 

English  of  the  XlVth  Century  ...... 

Stories  of  the  Old  World  ........ 

(Classics  for  Children.) 

English  of  Shakespeare    ......... 

.    .  lie-own  If  (Translation.)      ......... 

Sharp:  Be6wul£  (Text  and  Glossary) 
.    .  Harvard  Edition  of  Shakespeare  :  — 

20  Vol.  Edition.     Cloth,  retail    ..... 

10  Vol.  Edition.     Cloth,  retail    ..... 

Life,  Art,  and  Characters  of  Shakespeare. 
2  vols.    Cloth,  retail  .......    . 

New  School  Shakespeare.   Cloth.  Each  play 
Old  School  Shakespeare,  per  play     .    . 
Expurgated  Family  Shakespeare  ... 
Essays  on  Education,  English  Studies,  etc. 
Three  Vol.  Shakespeare,  per  vol  ...... 

Text-Book  of  Poetry     .......... 

Text-Book  of  Prose  ........... 

Pamphlet  Selections,  Prose  and  Poetry  .    . 
Classical  English  Header     ........ 

Merchant  of  Venice  ........... 

(Classics  for  Children.) 

Exodus  and  Daniel    ........... 

Robinson  Crusoe    ........... 

(Classics  for  Children.) 
Memory  Gems     ............. 

Chaucer's  Parlameut  of  Foules  ...... 

Manual  of  English  Prose  Literature 

Selections  from  Irving  |  Boards   \    '.'.'.. 
Two  Books  of  Paradise  Lost,  and  Lycidas  . 
Two  Shakespeare  Examinations    ..... 

Scott's  Quentin  Durward     ........ 

(Classics  for  Children.) 


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